11x11 Queens Puzzle #25
Difficulty: hard. Place 11 queens on this 11x11 board so that no two share a row, column or color region and none touch — including diagonally. This puzzle has exactly one solution, and it has been verified solvable by deduction alone: the complete 22-step reasoning is written out below.
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Step-by-step solution for puzzle #25
What follows is a complete solve of puzzle #25, worked the way a person works it rather than the way a computer does. It runs to 22 deductions — 11 queen placements and 11 elimination steps that narrow the board between them. Each step names the rule that forces it, so nothing below is a guess, and the hardest idea required is the pairing argument on columns.
Techniques this board needs: region shadow, pairing argument on rows, pairing argument on columns, last square in a region, region confined to one line.
Show the full walkthrough (14 steps) — contains the solution
Step 1. Wherever indigo ends up among its 3 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 2. Amber is down to 5 candidates in rows 1 and 2, columns 2–5, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about amber itself.
Step 3. The mint region has 3 squares left, spread across rows 3 and 4, columns 10 and 11. You do not need to know which of them takes the queen: every one of those options attacks the same 1 square, so they can be crossed out now.
Step 4. Wherever sea green ends up among its 7 squares, the same squares fall within reach of it — 1 square in total, impossible regardless of how the colour resolves.
Step 5. Indigo and amber are now boxed into rows 1 and 2 and nothing else. That is 2 colours needing 2 rows, so they fill those rows between them — which locks every remaining colour out and removes 10 squares at once.
The board after step 5 — 0 of 11 queens placed. Step 6. Here is the pairing argument. The blue and mint regions have no open squares outside rows 3 and 4 — 2 colours confined to 2 rows. Between them they must take one row each, in some order, so every other colour is shut out of all of them: 13 squares come off in one stroke.
Step 7. Indigo and mint are now boxed into columns 10 and 11 and nothing else. That is 2 colours needing 2 columns, so they fill those columns between them — which locks every remaining colour out and removes 14 squares at once.
Step 8. Pale yellow is down to 5 candidates in rows 5–8, columns 8 and 9, and they overlap enough to cover 2 squares elsewhere. Cross those out without deciding anything about pale yellow itself.
Step 9. Only row 9, column 9 is still open in orange, and since each colour must take a queen, that settles it. That clears row 9, column 9, the rest of the orange region and every square adjacent to it — diagonals included.
The board after step 9 — 1 of 11 queens placed. Step 10. The pale yellow region is down to one open square, at row 7, column 8. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 7 and column 8, the remaining pale yellow squares, and the ring of neighbours around row 7, column 8.
Step 11. Red is down to 3 candidates in rows 5, 6, and 8, columns 5–7, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
Step 12. All of the sea green region's remaining squares (2 open squares) lie in row 10. Its queen therefore has to be somewhere on that row, which means no other colour may use it: 7 squares come off.
Step 13. Wherever teal ends up among its 2 squares, the same squares fall within reach of it — 2 squares in total, impossible regardless of how the colour resolves.
Step 14. From here the board cascades. Keep alternating the region sweep and the line sweep and 9 queens fall out in order without any new idea being needed: sea green has one square left, so its queen takes row 10, column 7; red has one square left, so its queen takes row 6, column 6; teal has one square left, so its queen takes row 11, column 5; blue has one square left, so its queen takes row 3, column 4; amber has one square left, so its queen takes row 2, column 2; mint has one square left, so its queen takes row 4, column 11; indigo has one square left, so its queen takes row 1, column 10; khaki has one square left, so its queen takes row 8, column 1; slate has one square left, so its queen takes row 5, column 3. After each one, clear that row, that column, the rest of the colour and the ring of neighbours before looking for the next. The last of them seats the queen on row 5, column 3.
The board after step 14 — 11 of 11 queens placed.
That completes the board: 11 queens seated, one in every row, one in every column and one in every colour, with no two of them touching. Notice that nothing in the sequence above required a guess — each placement followed from an elimination already on the grid, which is what “single solution, solvable by logic” actually means in practice.
About 11x11 Queens puzzle #25
Puzzle #25 is one of the hard-rated boards in the 11x11 set. Eleven colors divide its 121 squares, and you win by placing eleven queens so that each row, each column and each color holds exactly one, with no two queens adjacent in any direction.
Like every board on this site, #25 has been verified to have exactly one valid solution, and that solution is reachable by deduction alone. You never have to guess and then unwind; every queen can be justified by something already visible on the grid. What changes from puzzle to puzzle is where the first certainty appears and how long the chain of reasoning runs before the next one — and that is entirely a product of how these particular colors are shaped.
How the colors divide this board
From smallest to largest, the regions on puzzle #25 are: indigo (3 squares), mint (4 squares), amber (5 squares), sea green (7 squares), pale yellow (11 squares), khaki (11 squares), teal (11 squares), orange (13 squares), blue (14 squares), red (15 squares), slate (27 squares). An even split would hand every color 11 squares, so the indigo region is running 8 squares under that average while the slate region is the roomiest area on the grid with 27 squares.
Why does the split matter? Because a region's square count has no bearing on how many queens it gets — the answer is always one. So indigo, with 3 squares, is under far more pressure than slate with 27. A 24-square gap like this one rewards players who start where the choices are fewest.
Shape matters as much as size. The indigo region is the most compact color here, filling 75% of the 2×2 box that encloses it, while teal is the most scattered at 46% of its 4×6 box. Compact regions are easy to eliminate around; straggling ones reach into more rows and columns and tend to stay unresolved until late.
Region by region
Each region's exact extent on this grid is listed below, starting with the most constrained. Knowing which lines a color can and cannot reach is most of the work on a hard board.
Indigo covers 3 squares, spanning 2 rows and 2 columns (rows 1–2, columns 10–11).
Mint occupies 4 squares within rows 2–4, columns 10–11.
Amber: 5 squares across rows 1–2, columns 2–5.
Sea green covers 7 squares, spanning 3 rows and 5 columns (rows 9–11, columns 6–10).
Pale yellow occupies 11 squares within rows 3–8, columns 8–10.
Khaki: 11 squares across rows 7–11, columns 1–4.
Teal covers 11 squares, spanning 4 rows and 6 columns (rows 8–11, columns 2–7).
Orange occupies 13 squares within rows 5–11, columns 9–11.
Blue: 14 squares across rows 1–4, columns 4–9.
Red covers 15 squares, spanning 7 rows and 4 columns (rows 3–9, columns 5–8).
Slate occupies 27 squares within rows 1–9, columns 1–6.
Where to start on this board
Indigo is the natural opening on #25: it is the smallest region on the grid at 3 squares, boxed into rows 1–2 and columns 10–11. Since one of those squares must hold a queen, that block of 2 rows and 2 columns is the most contested part of the board.
Indigo does not collapse onto a single line, so it will not hand you a queen on move one. What it does give you is a short list of 3 candidates. Take each one in turn, imagine the queen there, cross out its row, its column and the eight squares around it, and ask whether any other region is left with nowhere to go. One or two candidates usually fail that test immediately.
Work outward to the other flat regions next. Amber is no more than two rows tall, so its queen is confined to a narrow band. When two flat regions overlap the same pair of rows, those rows are spoken for and every other color is locked out of both — the single most productive pattern in the game, and it is live on this board.
The tightest lines on puzzle #25
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 5, 6, 10 and 11 are the narrowest, touched by only 4 colors. Column 1 is the tightest column, with 2 colors.
Count colors per line and you have a ranking of where to look. The narrowest rows here see only 4 colors, so one of that handful must take the row and the rest are squeezed into the other 10 rows. Row 9 sits at the far end with 6 colors crossing it; it will almost certainly be among the last to settle.
Working through the middle game
Past the opening, treat this as an elimination exercise rather than a placement exercise. Each queen takes out a row, a column and its eight neighbours, and the value of a move is measured by how much it removes. Cross everything out as you go and let the board do the remembering.
Keep alternating between two views of the board. In the region view you ask which of the eleven colors is down to one open square. In the line view you ask the same of each row and column. Both produce forced queens, and each placement refreshes the other view, so cycling between them deliberately is how a 11x11 board unwinds.
Say the reason out loud: "this region has one square left", "this column has one square left". If no such sentence is available, the eliminating is not finished. One more pass of X marks is almost always quicker than committing to a placement you cannot defend.
Why puzzle #25 is rated hard
The hard label comes out of four measurable things about #25: it has no region locked to one row or column, its smallest color spans 3 squares, its least crowded row meets 4 colors, and its largest and smallest regions differ by 24. Boards that score tight on those measures give up their first queen fast.
Nothing about the rating involves computation or chance. Each grid has a single solution reachable by pure logic; the rating simply measures how far ahead you have to reason. Easy boards resolve a queen every step or two, while hard ones can ask you to carry three unfinished lines of thought before any of them pays off.
If #25 feels harder than its rating suggests, the usual culprit is incomplete marking rather than a difficult deduction. Check that every queen already on the grid has its full cross of eliminations recorded — including the diagonal neighbours, which are by far the easiest to miss. More stalled solves come from one forgotten diagonal X than from any genuinely deep chain of logic.
Techniques that work well at 11x11
At 11x11 you are managing 121 squares, and holding the whole board in your head will not work. Split it into quadrants and solve locally. Most useful deductions on a grid this large are regional — a color pinched between two placed queens, or a band of rows that two regions must share. Trust the marks on the board over your memory.
Learn the pairing argument and you will use it on every board, this one included. If two colors can only sit inside the same two rows, between them they consume both rows and every other color is shut out. It generalises to any matching count — three colors across three rows, four across four. On #25, start hunting for it near column 1.
Use this as a rough check on your progress: the completed board has 4 queens around the border and 7 queens in the middle, and nothing in the corners. Solutions spread out; if yours is bunching up in one region of the grid, retrace a few moves.
How to play Queens
- The board is an N×N grid divided into N colored regions.
- Place exactly one queen (♛) in each row, each column, and each colored region.
- No two queens may touch each other — not horizontally, vertically, or diagonally.
- Tap once to mark a square with an X (where a queen can't go), tap again to place a queen, and once more to clear it.
- There is always exactly one valid solution. Use logic, never guessing, to find it.
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Questions about this puzzle
How hard is 11x11 Queens puzzle #25?
Puzzle #25 is rated hard. Its smallest color region holds 3 squares, its tightest row is touched by 4 colors, and no region is confined to a single row or column, so the opening move takes some hunting.
Does 11x11 puzzle #25 have only one solution?
Yes. Every puzzle in the 11x11 collection, including #25, has been verified to have exactly one valid arrangement of eleven queens, and it can always be reached by logic alone — no guessing or backtracking needed.
Where should I place my first queen on puzzle #25?
Start with the indigo region — the most constrained color on this board, with 3 squares spanning rows 1–2 and columns 10–11. Narrowing the smallest region first removes the most squares elsewhere on the grid.